**0**

votes

**0**answers

154 views

### On unique solutions to linear diophantine equations

Let $\sum_{i=1}^k a_ix_i = N$ be the equation with $a_i \in [2^t,2^{t+1}]$ being distinct primes. If we seek unique solutions $x_i\in R_i = (0,a_i)\cap \Bbb Z$, then in general it is not possible.
...

**5**

votes

**2**answers

1k views

### Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

This question has been asked previously on math.SE without receiving any answers.
http://math.stackexchange.com/questions/479740/solving-xkx1kx2k-cdotsxk-1k-xkk-for-k-in-mathbb-n
Letting $k$ be a ...

**1**

vote

**0**answers

238 views

### Can six square numbers be simultaneously represented in a single sum of consecutive odd numbers? [closed]

I had some free time from my work to do a little exploration regarding the existence (or non existence) of perfect cuboids. A solution is represented by the set of Diophantine equations:
$a^2 + b^2 = ...

**3**

votes

**0**answers

140 views

### Effective Lang-Weil bounds for del Pezzo surfaces

Let $X$ be variety in $\mathbb{P}^N$ over $\mathbb{F}_q$ of dimension $n$ and degree $d$.
By the Lang-Weil bounds
$ |\# X(\mathbb{F}_q) - q^n| \le (d-1)(d-2)q^{n-1/2} + Cq^{n-1}$for a constant $C$ ...

**2**

votes

**1**answer

291 views

### Are all sums of subsets of roots of unity unique?

For a prime $p$, let $S$ be the set of all $p$-roots of unity in the complex plane. Now, consider the sum, $W(R)$ of the members of a set $R$ which is a proper subset of $S$. I suspect that $R \ne R'$ ...

**14**

votes

**3**answers

2k views

### Which integers can be expressed as a sum of three cubes in infinitely many ways?

For fixed $n \in \mathbb{N}$ consider integer solutions to
$$x^3+y^3+z^3=n \qquad (1) $$
If $n$ is a cube or twice a cube, identities exist.
Elkies suggests no other polynomial identities are known.
...

**2**

votes

**3**answers

619 views

### Algorithm to count number of positive integer solutions of $x^2(8x-3)=y^2z$?

Given the Diophantine equation
$$ x^2(8x-3)=y^2z, $$
is there a way to efficiently count the number of solutions that satisfy $x+y+z\leq n$, where $n$ is a fixed given integer?
Also, for any fixed ...

**2**

votes

**3**answers

282 views

### Bounds on solutions to Diophantine equations of the form p(x)=q(y)

Thanks to responses in a previous question I asked, I was able to show that if $p$ and $q$ are two distinct polynomials with equal degree greater than 2 and coefficients in $\mathbb{N}$ that the ...

**2**

votes

**2**answers

268 views

### Four-Square Theorem for Negative Coefficient

What integers are not in the range of $a^2+b^2+c^2-x^2$ (for all integer combinations of a, b, c, and x)? This form is similar to that of Lagrange's Four-Square Theorem, for which the answer would be ...

**27**

votes

**6**answers

1k views

### Patterns among integer-distance points

Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.
...

**8**

votes

**1**answer

518 views

### Fundamental units of imaginary quartic fields

Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the ...

**0**

votes

**1**answer

550 views

### Is surface $x^2+z^2=2\cdot y^2$ something of a Möbius strip?

This question is naive. My association with Möbius strip comes from not being able to smoothly extract positive solutions of the diophantine equation
$$x^2+z^2=2\cdot y^2$$
I got a parametrization ...

**2**

votes

**0**answers

239 views

### Algorithm for solutions to quadratic forms over number fields

Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)?
I am especially interested in the quaternary case. There exist some ...

**5**

votes

**0**answers

203 views

### When does the Lloyd polynomial have only integral roots?

For a $t$-error correcting code of length $n$ over the finite field $\mathbb{F}_q$, the Lloyd polynomial is given by
$$
L_t(n,x):=\sum_{j=0}^t(-1)^j\binom{x-1}{j}\binom{n-x}{t-j}(q-1)^{t-j}.
$$
A ...

**9**

votes

**3**answers

2k views

### The modular arithmetic contradiction trick for Diophantine equations

It is a slick, and seemingly ad-hoc, technique often used to prove that a Diophantine equation has no solutions.
The equation $f(x_1,\ldots, x_k)=0$, with variables $x_i\in\mathbb{Z}$ and some ...

**0**

votes

**0**answers

1k views

### Diophantine: a^n + b^n + c^n = d^n and a^n + b^n = c^n + d^n

Let us consider the equation $a^n+b^n=c^n$ for positive integers $a,b,c$ and $n\ge 2$. The $n=2$ case has a well-known and beautiful parametrization known as Pythagorean triples. Fermat's Last Theorem ...

**20**

votes

**1**answer

726 views

### Is there an online encyclopedia of Diophantine equations (OEDE)?

Hello all!
I'm just wondering if there is an online encyclopedia of Diophantine equations (OEDE), analogous to the OEIS for sequences.
While trying to solve one Diophantine equation, I reduced the ...

**0**

votes

**2**answers

381 views

### Classification of these Binary Quadratic Forms

What are necessary and sufficient conditions on a binary quadratic form $ax^2+bxy+cy^2$, with integer coefficients and solution set in integers, to be equivalent to $x^2-y^2$, and separately to ...

**4**

votes

**3**answers

614 views

### Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares?
Be it 1,2,3,...n or k,k+1,k+2,...n.

**1**

vote

**1**answer

1k views

### The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial.
In particular, it's the case ...

**6**

votes

**4**answers

916 views

### Can repunits be perfect cubes?

Is it true that the equation $10^{n}-9m^{3}=1$ has only one positive integer solution, namely $n=m=1$? I can't find the answer. This has an equivalent description that the repunits $R_n = 11\dots1$ ...

**2**

votes

**2**answers

274 views

### Help with this system of Diophantine equations

A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it.
This is the actual question which I'm trying to solve:-
For ...

**1**

vote

**3**answers

328 views

### Help with this Diophantine equation

Note: This question was posted in error, and should be closed as no longer relevant. The correct question is posted at Help with this system of Diophantine equations (End of note)
For a research ...

**10**

votes

**1**answer

1k views

### Effect of abc conjecture on Fermat's Last Theorem

A website ( http://www.math.unicaen.fr/~nitaj/abc.html#Consequences ) says that the $abc$ conjecture implies that there are only finitely many solutions to the equation $x^n+y^n=z^n$ with ...

**3**

votes

**1**answer

224 views

### Diophantine equation with primitive nth root of unity

Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - ...

**2**

votes

**0**answers

142 views

### Reference for original paper (but translated to English) of Matiyasevich's proof of Fibonacci relation being Diophantine?

Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems that is accessible to ...

**12**

votes

**2**answers

305 views

### A sequence based on Catalan–Mihăilescu problem

It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The conjecture was ...

**3**

votes

**2**answers

1k views

### What is known about a^2 + b^2 = c^2 + d^2

Could you state or direct me to results regarding the Diophantine equation $a^2+b^2=c^2+d^2$ over integers? Specifically, I am looking for a complete parametrization. In the case that a complete ...

**18**

votes

**1**answer

934 views

### Solutions to $\binom{n}{5} = 2 \binom{m}{5}$

In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says:
On National Public Radio, the Weekend Edition program posed the
following probability problem: Given a certain number of ...

**6**

votes

**3**answers

830 views

### References on techniques for solving equations with discontinuous functions such as floor and ceiling?

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good ...

**4**

votes

**0**answers

318 views

### $a^5+b^5=c^5+d^5$ and polynomial identities

No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from genus 0 curve ...

**5**

votes

**1**answer

305 views

### What analytic tools can provide a lower bound for this Diophantine equation?

The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken.
I felt that the following ...

**10**

votes

**1**answer

314 views

### Binary expansion of squares

I came across the following simple question: what odd integer squares have exactly 3 ones in their binary expansion?
After looking at it for a while I convinced myself that the only solutions to $r^2 ...

**12**

votes

**2**answers

961 views

### sum of three cubes and parametric solutions

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples ...

**3**

votes

**2**answers

256 views

### Catalan-type equations for prime powers

Do there exist nonzero integers $a,b,c$ for which the equation $$aX + bY = cZ$$ has infinitely many solutions with $X,Y,Z$ distinct prime powers?
For example, if there are infinitely many Sophie ...

**8**

votes

**1**answer

309 views

### When adding a constant makes a multivariate polynomial reducible?

Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of smaller degrees?
...

**5**

votes

**2**answers

256 views

### Semimagic Squares and Partitions

Say, we have a semimagic square $X$, that is, an $n\times n$ square matrix with entries from natural numbers, such that each row and column of it sums up to the same natural number $s$.
Let $M$ be a ...

**0**

votes

**1**answer

441 views

### Like Diophantine equation

Dear all,
I have posted this question on m.s.e. Unfortunately, no one responded to answer. I hope, this site and members of this site will answer my questions.
The equation $x^n - ny^x-nxy$ = $0$ ...

**4**

votes

**0**answers

240 views

### Expressions of $tanh$ type whose continued fractions have two shifts per period

This is a follow-up of another thread about quasi periodic continued fractions, a.k.a. Hurwitz fractions, with some linear shifts. I seem to have found the pattern of a subclass of them, as given ...

**5**

votes

**6**answers

649 views

### Representations with Triangular Numbers

A well known theorem of Gauss says that any natural number $n$ may
be written as the sum of three triangular numbers -
$$
n={a_{1} \choose 2}+{a_{2} \choose 2}+{a_{3} \choose 2}
$$
The following ...

**10**

votes

**1**answer

276 views

### Sets of integers represented by degree zero rational functions

Suppose $f(x_1,x_2,\dots)=\frac{P}{Q}$, where $P,Q$ are polynomials in several variables with integer coefficients that have the same degree. Let's denote by $S(f)$ the set of integers $n$ for which ...

**6**

votes

**4**answers

1k views

### Does every polynomial diophantine equation have solutions modulo p?

Obviously, this is not exactly true; what I am really asking is whether any diophantine polynomial equation with integer coefficients (let's call them DPEICs) who's solution does not admit ...

**3**

votes

**0**answers

272 views

### quasi periodic continued fractions and powers of e, tanh, tan

It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N $, when written as regular continued fractions (R.C.F.), yield what can be called a ...

**4**

votes

**2**answers

495 views

### Are there Heronian triangles that can be decomposed into three smaller ones?

Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...

**3**

votes

**1**answer

218 views

### Any non-conforming numbers?

Consider the function $x^m \pm y^n \pm z^p$, where $x, y, z, m, n, p$ are integers such that $m, n, p \geq 2$. The question is, are all numbers expressable using this function? Are there any ...

**3**

votes

**0**answers

95 views

### Weak classes of diophantine functions

From a well-known work(s) by Putnam, Davis, Robinson and Matiyasevich, we know that every partially recursive function is diophantine.
Now it seems a natural question to ask: can we say something ...

**0**

votes

**0**answers

337 views

### Diophantine equation solutions

I am not able to make headway on solving the diophantine equation $x^m - y^n = 6.$ Are there any solutions to this? What about $x^m - y^n = 14,$ and $= 30$ (both $m$ and $n$ are at least $2$).

**2**

votes

**2**answers

248 views

### Maximum size of powers with a given difference

Pillai's conjecture -- that the gap between (nontrivial) powers is unbounded below -- is still open (it would be a consequence of the $abc$ conjecture, were that proven). But I wonder what the right ...

**42**

votes

**3**answers

2k views

### What is the geometry of an undecidable diophantine equation?

As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine ...

**5**

votes

**2**answers

888 views

### The diophantine equation X^2 - Y^2 - Z^2 = +- 1

Hi everybody. I'd like to know if the diophantine equation
(1) $$X^2 - Y^2 - Z^2 = \pm 1$$
has been studied and if the set of its solutions $(X,Y,Z)$ is known. I appreciate any reference. Thank you ...